Matrix multiplication

• Matrix multiplication is not commutative; in general, the order in which matrices are multiplied matters. In other words, for two matrices A and B, it’s not always the case that AB = BA.

• Before multiplying matrices, ensure they conform to the multiplication rule: the number of columns in the first matrix must match the number of rows in the second matrix.

• Multiplying a matrix by a scalar involves multiplying each entry of the matrix by the scalar.

• When multiplying two matrices, the element in the i-th row and j-th column of the resultant matrix will be the dot product of the i-th row of the first matrix and the j-th column of the second matrix.

• Multiplication of matrices is associative. This means for three matrices A, B, and C, the product A(BC) is the same as the product (AB)C.

• Matrix multiplication distributes over matrix addition. For three matrices (A, B, and C) and two matrices (A and B) with the same dimensions, we have A(B + C) = AB + AC and (A + B)C = AC + BC.

• The multiplication of any matrix by the identity matrix (a matrix with ones on the diagonal and zeros elsewhere) results in the original matrix. That is, if A is a matrix and I is the identity matrix, then AI = IA = A.

• In some instances, multiplying a matrix by another matrix gives the zero matrix. The notion of a ‘zero-divisor’ is important and can lead to broader discussions and studies about matrix invertibility and determinants.

• Matrix multiplication is used in a variety of mathematical contexts, including transformations in geometry, solving systems of simultaneous equations, and representing and analyzing networks in graph theory.

Review these points and use practice problems to solidify your understanding and ability to apply these concepts. Matrix multiplication, while complex, is a powerful tool in pure mathematics and many other fields.